Wednesday, March 2, 2011

FORCING ENTRY

Hello, world.

I should begin with a quick word about what this blog is. The Form & Formalism Working Group began in November, 2009, in the wake the first annual "Form & Formalism" conference, held at the Jan Van Eyck Academie in Maastricht, and orchestrated by Tzuchien Tho of the Versus Laboratory research project. A second conference followed in 2010, and Versus is in the process of planning a third for the coming Fall. (Programmes for both FF conferences can be found here: http://versuslaboratory.janvaneyck.nl/events/view/5 and here: http://versuslaboratory.janvaneyck.nl/events/view/11.) From the conferences formed the group, and from the group now comes the blog. Nothing else needs to be said about this just yet.

To get the ball rolling, I've decided to make available here a few short texts that I've been working on, still in a somewhat rough state, for the Badiou Dictionary that Steve Corcoran is in the process of pulling together for Edinburgh University Press. Your comments, corrections, criticism, etc. are of course welcome.

I'll try to post an entry every day or so over the next week. Today, FORCING. Stay tuned for GENERIC, MODEL, SUTURE, IDEOLOGY, ONE, and VOID.
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DEFINITION OF FORCING

‘Forcing’ (forçage) is among Badiou’s signature expressions. From 1968 onwards, it has traced out an axial category in his work. Despite certain shifts in meaning, it has sustained a cluster of ideas and concerns that are, throughout, invariant. The core idea seems to be this: what Badiou calls ‘forcing’ is in each case a radical and systematic transformation of a situation by means of series of actions acting upon, or proceeding from, the real of the situation—that which, prior to the activity of forcing, subsists in the situation as an invisible, unoccupiable, or ‘impossible’ site, occluded by knowledge and cloaked by (the dominant) ideology. Invariant too is that it is in each case mathematics that conditions the category of forcing.

We can split Badiou’s use of the term into two periods, each of which presents a crucial variation on this central theme. The pivotal texts of each period, in which the concept of forcing is formulated or reformulated, are ‘Infinitesimal Subversion’ and Being and Event. (For completeness’ sake, Theory of the Subject should also be mentioned, but despite the importance that a whole array of related and analogous concepts of force, torsion, etc. play in that text, the idea of forcing, itself, appears only in a transitional capacity.)

Forcing in ‘Infinitesimal Subversion’

The context of ‘Infinitesimal Subversion’ is the project undertaken by several members of Le Cercle d’épistémologie—the working group behind Les Cahiers pour l’Analyse, and to which Badiou belonged in the last years of the 1960s—to develop a general theory of structural change, informed by both Lacanian psychoanalysis and Althusserian historical materialism—a project which gave Badiou’s enterprise its initial and lasting coordinates. With minimal violence, we can characterize it according to the following theses, which find their canonical expression in Jacques-Alain Miller’s ‘The Action of Structure’ (published alongside ‘Infinitesimal Subversion’ in Cahiers vol. 9):

1.The structure of a situation always has at least one ‘empty place’, a place which cannot, according to the structure, be occupied. It is characterized by a certain structural impossibility, as the ‘Real’ of the situation. Jacques-Alain Miller calls this the ‘utopic point’of the structure (97).

2.The empty place is, in general, indiscernible. It is a ‘blind spot’, unstably masked or ‘sutured’ by ideological or imaginary illusion.

3.“Any activity which does not play itself out entirely in the imaginary but which is to transform the state of the structure,” Miller writes, “departs from the utopic point, the strategic post,” specific to the situation. (97)

If the reader of these remarks recognizes in the idea of a ‘utopic point’, not only an echo of Lévi-Strauss’ ‘floating signifier’ and a fellow traveler of Deleuze’s ‘empty square’ of structure, but a prefiguration of Badiou’s later category of the ‘evental site’, then she is on the right track. But first we must turn to the text where the idea of transforming a situation from the bias of its utopic point is first thought through under the condition of mathematics, for it by placing this notion under the mathematical condition that the category of forcing is won.

This all gets underway in ‘Infinitesimal Subversion’, where Badiou transports Miller’s schema into the laboratory of formal mathematics and model theory, in an analysis of Abraham Robinson’s invention of non-standard analysis. The situation’s structured space of possibilities here becomes the space of inscriptions allowed by a formal axiomatic—the formulae that it can demonstrate. The Real, the ‘utopic point’, becomes the place of the underivable, the space unoccupiable by formulae licensed by the formalism. That every consistent formalism is punctuated by such impossibilities is an iron necessity; if every place could be occupied, and every expressible formula written down as a theorem, formalism would become “an opaque body, a deregulated grammar, a language thick with nothing” (SI 122), which is to say, inconsistent. A formalism is only consistent—and so, in virtue of Gödel’s completeness theorem, interpretable in a model—if there is at least one formula which it can express but not demonstrate; it is “owing to the exclusion of certain statements, the impossibility of having the constants occupy certain constructible places, that an axiomatic system can operate as the system it is, and allow itself to be thought differentially as the discourse of a real” (122).

The demarcation of an unoccupiable place is made precise in mathematics with its syntactical distinction between constants and variables. The system of finite arithmetic, for instance, allows no constant—no integer—to be substituted for the y in the expression ‘For all integers n, n ≤ y’—but by recourse to the variable ‘y’ it is able to mark this inoccupiable place, without, for all that, occupying it.[1] “A variable,” Badiou writes, “ensures that impossible equations are sufficiently legible to read their impossibility;” it is the "operator of the real for a domain, it in fact authorizes within that domain the writing of the impossible proper to it. The existent has as its category a being-able-not-to-be the value of a variable at the place it marks." (SI, 122)

‘Forcing’ is a procedure of radically transforming the structure by occupying one or more ofitsreal, unoccupiable places, without for all that collapsing the structure into sheer inconsistency. It begins with an act of nomination, the definition of a constant that occupies an inoccupiable place, closing one of the open and formerly unsatisfiable sentences in which only variables could once be written. Robinson’s intervention consists in defining a new constant a and axiomatically stipulating it to be such that for all real numbers n, n ≤ a, a gesture which, by occupying the inoccupiable, marks an intrusion of formalization into the real that was its impasse.[2] (Badiou calls occupation the inscription of an ‘infinity-point’, though the general concept is meant to apply to constants like the ‘imaginary number’ with which Bombelli breached the x in ‘x2 + 1= 0’, which the existing algebra had declared inoccupiable.) The forcing procedure continues with a submission of the new constant to all the remaining operations of the initial system—a, for instance, can be added to, divided by, and so on, and so Robinson is able to define infinitesimals simply as multiples of 1/a. In sum,

the infinity-point is the marking of something inaccessible for the domain; a marking completed by a forcing of procedures, constraining them to be applied to precisely that which they had excluded. Of course, this forcing entails a modification of the way in which the domain is set out, since the constructible objects in the higher domain are able to occupy places which those of the domain itself ‘inoccupy’. The new space in which the procedures can be exercised is disconnected from that which preceded it. The models of the system are stratified. (SI 120)

This brings about not merely an extension, but a transformation of the domain in question: new patterns are unleashed, old ones often destroyed. And so Badiou goes on to identify this forcing procedure as a “reforging” (refonte) of the structure, connecting it explicitly with the theory of epistemological breaks, a theory which he inherits, with modifications, from Althusser and Bachelard—a recognizable prototype of the theory of truths unleashed in Being and Event.[3]

Forcing in Being and Event

Between the theory of forcing presented in ‘Infinitesimal Subversion’ and the one we find in Being and Event, intervenes the a new and decisive condition: a technique developed by Paul J. Cohen in his proof of the independence of the Generalized Continuum Hypothesis and the Axiom of Choice from the axioms of Zermelo-Fraenkel set theory (ZF), which likewise appears under the name of ‘forcing’. Before we address its incorporation into Badiou’s philosophical apparatus, we will take a quick look at forcing in its native, mathematical terrain.

It is, once again, set-theoretical model theory that provides Badiou with the requisite conceptual (scientific) material. Like Robinson’s procedure for the making of ‘non-standard’ models, Cohen’s forcing technique is, at bottom, a systematic way of generating a new model from a model already given. The main thrust of Cohen’s proof is to take a countable, transitive model[4] of ZF and ‘force’ the existence of a new model by supplementing it with a generic element included in, but not belonging to, to initial model—together with all the sets which can be constructed on the supplement’s basis by licensed by the ZF axiomatic. Considered in its logical structure, forcing is a relation of the form ‘a forces P’, where a is a set and P a proposition that will hold in the generic extension of the initial model—provided thata turns out to belongto the generic supplement on which that extension is based. In this respect, forcing resembles a logical inference relation, but one that differs markedly from the inference relation of classical logic—the law of the excluded middle, in particular, does not hold for the forcing relation, and the logic it generates is essentially intuitionistic.[5]

As Cohen has shown, the consequences this supplementation can be quite extraordinary, and go far beyond simply adding a new set’s name to the census. The generic supplement, for instance, may be structured so as to induce a one-to-one correspondence between transfinite ordinals that, in the initial model/situation, counted as distinct orders of infinity, thereby collapsing them onto one another and making them effectively equal. Cohen exploited this possibility to great effect by taking the model that Gödel had built in order to show that the Generalized Continuum Hypothesis (GCH)—the thesis that the size of the set of subsets set of any transfinite cardinal number Àn is equal in size to the next greatest cardinal Àn+1—is consistent with ZF (a model in which the continuum hypothesis holds), and on its basis forcing a generic extension in which the continuum hypothesis fails (the extension being a model in which the set of subsets of Àn is demonstrably equal to almost any cardinal whatsoever, so long as it’s larger than Àn), thereby demonstrating the consistency of GCH’s negation with the theory, and hence the independence, or undecidability, of GCH with respect to ZF.

Being and Event recovers Cohen’s concept and enlists it in a re-articulation of the existing category of forcing: the set underlying the model is now seized upon as the situation that forcing will transform, and faithful Miller’s cartography of change, Badiou adds that the whole procedure—both the articulation of the generic truth and the forcing of its consequences for the situation to come—must in every case proceed from an anomalous occurrence in the ‘utopic point’ of the situation in question, now rechristened ‘evental site’. Though it is now Cohen rather than Robinson whose mathematics condition Badiou’s theory of change, the new category of forcing preserves most of the features familiar to us from “Infinitesimal Subversion.” One crucial difference, however, is that the whole process is now seized as a logic of subjective action: Forcing is now names “the law of the subject” [CITE], the form by which a subject faithful to an event transforms her situation into one to which a still-unknown[6]truth (understood as a generic subset of the initial situation) well and truly belongs, by deriving consequences that the inscription of this new constant will have brought about.

In light of Being and Event’s decision to interpret ZF as the theory of being qua being, and forcing as the form of a subject’s truth-bearing practice, Badiou extracts two lessons from Cohen’s proof of the undecidability of GCH: first, that it demonstrates the existence of a radical ontological gap or ‘impasse’ between infinite multiplicities and the sets of their subsets (to which Badiou associated the notions of ‘representation’ or ‘state of a situation’), the exact measure of which is indeterminate at the level of being-in-itself; second, that this ontological undecidability is nevertheless decidable in practice, but only through the faithful effectuation of a truth, suspended from the anomalous occurrence of an event.

[1] It goes without saying that variables can also be used to mark occupiable places.

[2] This is a bit of a simplification. Robinson’s procedure is carried out not only with respect to ≤ but for every relation R such that for every finite set of constants {a1, …, an}, there exists a y such that a1Ry & … & anRy. (Robinson calls these relations ‘concurrent’; Badiou, with typical pizzazz, calls them ‘transgressive’.) For each of these relations Ri, the idea is to introduce a new constantaiwhich is axiomatically stipulated to be such that yRiai for all y. Amongst these relations, and certainly the most interesting for the present case, is ≤.

[3] In describing this procedure, Robinson himself does not use the word ‘forcing’—which, at the time of his invention of non-standard analysis, had recently been given a technical coinage by Paul Cohen, whose ‘forcing’ concept would (by 1988) become a decisive condition for Badiou’s project of developing the philosophical category of forcing that we already see in motion here. Interestingly enough, a year or so after the publication of “Infinitesimal Subversion” in the Cahiers pour l’analyse, Robinson turned his attention to Cohen’s work and, on its basis, developed a new form of forcing, in Cohen’s sense, which is now known as ‘infinite model theoretic forcing’ or, simply, as ‘Robinson forcing’. Badiou, to my knowledge, has not commented on this device.

[4] A model structure M is transitive if x belongs to M whenever x belongs to y and y belongs to M (for all x, y). For a quick explanation of how ZF can have countable models, see the entry for model.

[5] For details, see Z.L. Fraser, “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus”, in The Praxis of Alain Badiou, Melbourne, re.press, 2007: pp. 23-70.

[6] According to Hao Wang, Kurt Gödel once described forcing as “a method to make true statements about something of which we know nothing.” (Hao Wang, Kurt Gödel: A Logical Journey, (Cambridge, Mass.: MIT Press, 1996), 252.)